Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

learn more… | top users | synonyms

2
votes
3answers
64 views

Simplify the expression $(2n)!/(2n+2)!$

I'm a little confused as to how $(2n)!/(2n+2)!$ looks when written out. Basically I'm trying to visualise it so that I know how to cancel this and like terms in future.
0
votes
0answers
34 views

lower bound for factorials product

For $n$ positive integer, let $F(n) = 1! × 2! × 3! × 4! × \cdots × n!$, product of factorial(i) for $i$ in $[1\ldots n]$. Let $G(n) = \{i \in [1\ldots n],\text{ such that }n\mid F(i)\}$. It is ...
1
vote
1answer
24 views

Factorials in Sigma Notation

Geometric Series one would use $S_n = \dfrac{a_1\cdot (1 - r^n)}{(1 -r)}$. Arithmetic Series one would use $S_n = \dfrac{n\cdot (a_1 + a_n)}{2}$. But how would I convert a sigma notation problem with ...
5
votes
2answers
48 views

The number of primes in the factorization of $N!$

Is there an approximation to the number of primes in the factorization of $N!$? For example: For $N=10$, this number is $15$. For $N=100$, this number is $239$. For $N=1000$, this number is $2877$. ...
1
vote
1answer
47 views

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$.

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$. I'm trying to prove the statement by building on my observation that $(1-\frac{1}{n})^n$ ...
1
vote
2answers
22 views

Factorial simplication question

How does the following: $$(k+1)! - 1 + (k+1).(k+1)!$$ simplify to: $$ (1+k+1).(k+1)! - 1 $$ and then $$(k+2)! - 1$$ I just can't seem to see how that works, I've tried writing out the factorials ...
0
votes
2answers
26 views

How can we prove this = 1 for all n

$\displaystyle n!-\sum_{k=1}^{n-1}k\cdot k!$ By computing this by hand for several small values of $n$ I can see that it is always equal to 1. But I can't see how to prove that.
1
vote
3answers
51 views

Factoring added factorials

How do I facilitate prime factorization without brute-forcing the 600+ digit number? For example, how would I factor (82! + 83! + 84!) ?
1
vote
1answer
58 views

Upper bound for $n!$

Let $a\in\mathbb{N}$. is there an upper bound be for the smallest n so that $n!>a$? It doesn't have to be a good upper bound, just something that works. Thanks.
0
votes
2answers
43 views

Number of primes in $[30! + 2, 30! + 30]$

How to find number of primes numbers $\pi(x)$ in $[30! + 2$ , $30! + 30]$, where $n!$ is defined as: $$n!= n(n-1)(n-2)\cdots3\times2\times1$$ Using Fermat's Theorem: $130=1\mod31$, (since $31 \in ...
1
vote
4answers
84 views

Demonstrate that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} \simeq 1.7 \sqrt{n}$

As in the title, I know that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} = \frac{(2n - 2)(2n - 4)\cdots 4 \cdot 2}{(2n - 3)(2n - 5) \cdots 3 \cdot 1} \simeq 1.7 \sqrt{n}$ Could you give some hint ...
0
votes
3answers
40 views

series calculation involving factorial

How would one calculate following $$\sum_{k=2}^\infty \frac{k^2+3k}{k!}$$ I searched youtube for tutorials (patricJMT and other sources) where I usually find answers for my math problems, I think I ...
-5
votes
1answer
62 views

Lottery Canada Statistics Lie?!?! [closed]

I'm not a statistics guru, but I took issue with a national lottery in Canada called 'Lotto Max'. LottoMax involves 49 different numbers (1 to 49). The odds of winning a prize are based on the ...
2
votes
0answers
47 views

How to prove these indentities?

How to prove these indentities? $\displaystyle \sum \limits_{k\geq0} {2n\choose 2k-1}{k-1\choose m-1}=2^{2n-2m+1}{2n-m\choose m-1}$ $\displaystyle \sum \limits_{k=0}^{m} {m\choose k}{n+k\choose ...
2
votes
1answer
33 views

Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
0
votes
2answers
33 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
3
votes
0answers
64 views

derangements and permutations in cryptography

i have a problem that i am having a bit of trouble with; we are given a partial key (missing 11 letters) for a mono-alphabetic substitution cipher and asked to calculate the number of possible keys ...
0
votes
1answer
16 views

Name of numbers in “to the power of” and factorial calculations

In $4*5=20$ , $4$ and $5$ are multiplicands and $20$ is the product. What are the names / labels of the numbers in the following expressions? $2^3=8$ $4!=24$
8
votes
3answers
119 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
1
vote
2answers
33 views

Can non-integer factorials be calculated without numerical integration?

I saw a strange way to write the factorial function somewhere and after some integration by parts, it all sure enough worked out. $$ n! = \int_0^\infty x^{n}e^{-x}dx $$ $$ ...
4
votes
1answer
61 views

Field Theory, Factor Ring, Polynomials

I have the following problems: (1) Let $g=X^2+\overline{4}$ and $h=X^2+\overline{2}$ be polynomials in $(\mathbb{Z}/\mathbb{Z}7)[X]$. $L$ and $K$ are the splitting fields of $g$ and $h$ over ...
2
votes
1answer
35 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
1
vote
3answers
245 views

Prove that no number in this list is prime - Formatting a proof advice

Question: Let $n \in \mathbb{Z}$ where $n \geq 2$, prove no number in the list: $$n! + 2, n! + 3,...,n! + n$$ is prime. I have written my proof exactly as follows: Proof: $P(n) = n! + n = ...
2
votes
1answer
45 views

Using combinatorial reasoning to show $n!=\binom{n}{0}D_n+\binom{n}{1}D_{n-1}+\dots+\binom{n}{n}D_0$

How can one use combinatorial reasoning to show that $$n!=\dbinom{n}{0}D_n+\dbinom{n}{1}D_{n-1}+\dbinom{n}{2}D_{n-2}+....+\dbinom{n}{n-1}D_1+\dbinom{n}{n}D_0$$ Now $D$ stands for deranged which is a ...
0
votes
7answers
138 views

Calculating $\binom{1}{2}$

Show $\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and ...
-1
votes
1answer
71 views

Relation/connection between $n!$ or $e$ and $2^n$

What is the relation/connection between $n!$ or $e$ and $2^n$ ? Is the there a relation/connection between $n!$ or $e$ and $2^n$?
9
votes
2answers
135 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
4
votes
3answers
104 views

Evaluating $\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$ [duplicate]

$$\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$$ Now, $\log(n!) = \Theta (n\log(n))$ so I think we could write, $$\lim_{n\to\infty}\frac{1}{2n}\left(\log\left(2n!\right) - ...
-2
votes
1answer
57 views

Determine Even or Odd

Let there a number $n$ and $b$. We need to tell if $n!$ divided by $b$ would be odd or even. How could we determine this problem. Assume if (n! % b)!=0 then answer gets rounded to integer.
6
votes
1answer
96 views

$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
-2
votes
2answers
99 views

Even or Odd for factorial

Moderator Note: This is a current contest question on codechef.com. Given $N$ and $M$ I need to tell whether $\left\lfloor \large\frac{N!}{M} \right\rfloor$ is even or odd.How to do this ...
0
votes
0answers
49 views

Smallest $n$ such that $n!>a^{x}$

Given some $a,x \in \mathbb{N}$, what is the smallest $n$ such that $n!>a^{x}$?
8
votes
1answer
134 views

Study of the convergence of a sequence with repeated radicals

Let the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } } $$ Does this sequence converge? I can tell intuitively that $a_n$ is monotonically increasing. Therefore, there are two ...
0
votes
1answer
66 views

Digit in units place of 1!+2!+…99!

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were 3, 0, 1 and 7. I am absolutely stumped. Any pointers? By ...
3
votes
0answers
49 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
0
votes
1answer
68 views

Falling factorial summands

Representing the summand as a falling factorial Compute the sum $$\sum_{k=1}^n\frac{1}{(k+1)(k+2)}$$
0
votes
2answers
86 views

Trailing zeroes in factorials: are there any excluded values divisible by 5 other than $5$ and $30$?

I've discovered that when this algorithm for counting zeroes on the end of $n!$ is applied to some $n\in\Bbb{N}$: $$f(n)=\sum_{k=1}^{k:n/5^k\le1}\left\lfloor\frac{n}{5^k}\right\rfloor\notin\{5,30\}$$ ...
1
vote
1answer
61 views

Stirling approximation / Gamma function

Is it possible to obtain the Stirling approximation for the factorial by using the gamma function ? $$\Gamma(z) = \lim_{n \to +\infty} \frac{n! n^{z}}{z(z+1) \dots (z+n)}$$ Any hint ?
8
votes
3answers
226 views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
3
votes
2answers
84 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
3
votes
2answers
99 views

Limit of the sequence $\frac {a^n} {n!}$

I need to prove that $$\lim_{n \rightarrow \infty} \frac {a^n} {n!}=0$$ I have no condition over $a$, just that is a real number. I thought of using L'Hôpital, but it's way too complicated for ...
1
vote
0answers
38 views

Approximating the probability that a range bounds a given number, with very large numbers

Let $m$ numbers be chosen uniformly from $0,\dots,n-1$ without replacement and then sorted in ascending order as $\ell_0,\dots,\ell_{m-1}$. Let there be $b,e,x$ such that $0 \le b \le e \le m$ and $0 ...
4
votes
3answers
156 views

How does $\tbinom{4n}{2n}$ relate to $\tbinom{2n}{n}$?

I got this question in my mind when I was working on a solution to factorial recurrence and came up with this recurrence relation: $$(2n)!=\binom{2n}{n}(n!)^2$$ which made me wonder: is there also a ...
2
votes
4answers
384 views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
0
votes
1answer
55 views

Closed form of $n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$

$n$ is given, and it takes part in the following formula. $$n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$$ Is there a nicer way for expressing it? Without the summation sign?
1
vote
1answer
75 views

What's a useful recurrence relation for $(2n)!$ in terms of $n!$?

I want to make an algorithm for $n!$ which will divide the number in half and call the algorithm again until n is so close to $0$ that a value of $1$ can be safely returned, and use the value of each ...
0
votes
1answer
88 views

Exotic 6-horse race betting probabilities

I'm gearing up for horse racing season, and I'm trying to teach some fellow engineering friends how to bet "exotic" bets by using colored dice to simulate horses. So, the odds for each horse winning ...
9
votes
3answers
1k views

Why factorials above 85 contain zero's at the end.

Sorry, I'm not into advanced math, but it wonders me, why factorials above ~85! contain lots of zero's at the end. Example, 100! = ...
3
votes
3answers
160 views

Does $n!$ divide $ n^n$?

Today while I was reading on how to shuffle an array I came across a statement that claims we shall not swap an array entry with the whole array range when shuffling the array otherwise we end up with ...
3
votes
3answers
73 views

N women and N men. Groups of pairs.

So we have $N$ men and $N$ women. We are creating groups of pairs. It is not necessary to use every man and woman. How many groups can we make ? So if we number them from $1$ to $N$ - let $W_{1}$ be ...